Nuclear overlap function

Once one has extracted T e and an explanation of the GP effect on the nuclear dynamics will follow immediately. The dynamics in T e is decoupled from the dynamics in To, and thus any observable will show GP effects only if the corresponding operator samples and To in a region of space where these functions overlap. In a nonencircling nuclear wave function, and never overlap, and this gives us a diagrammatic proof (Fig. 4) of the well-known result that a nonencircling wave functions shows no nontrivial GP effects. 

When the electron spins are coupled with nuclear spins, the cross relaxation accompanying the change of the nuclear spin state can occur. In this case the apparent spectral overlap of the A and the B spins is not necessary. The spectral averaging of Eq. (3) is therefore a difficult task. Instead, we assume that the spectral overlap function in Eq. (2) is given by a constant F. Then, the spatial averaging of Eq. (3) is necessary for correlating the observed relaxation kinetics with the theory. The result of the spatial averaging will be shown for the two extreme cases of the spatial distribution of radicals in solids. 

In certain cases, the classical Marcus formula is not sufficient to explain the observed-dependence of the electron transfer rate on temperature or AG, which could indicate that it is necessary to use a Franck-Condon term in which the contribution of the nuclei is treated in quantum mechanical terms. In this treatment, the Franck-Condon term equals the thermally-weighted sum of the contributions from all possible vibrational states of the reactants, each multiplied by their Franck-Condon factor i.e. the square of the overlap integral of a nuclear wave function of the reactant with the nuclear wave function of the product state that has the same total energy. 

Figure 13 Potential energy surfaces for electron transfer reactions. Hamionic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Namre Publishing Group, www.nature.com)...

In molecules and clusters, genuine exchange (as well as correlation) among identical nuclei is very small because, at typical internuclear separations, the overlap of nuclear wave functions is rather small. However, the exact xc functional also contains self-exchange contributions which are not small and which cancel the self-interaction terms contained in the Hartree potentials in Eqs. (71) and (72). Hence it will be a very good approximation to represent Fjc by the self-exchange terms alone. This leads to... 

Electronic motion with a typical frequency of 3 x 10 s" (i>= 10 cm ) is much faster than vibrational motion with a typical frequency of 3 x 10 s (v = 10 cm" ). As a result of this, the electric vector of light of frequencies appropriate for electronic excitation oscillates far too fast for the nuclei to follow it faithfully, so the wave function for the nuclear motion is still nearly the same immediately after the transition as before. The vibrational level of the excited state whose vibrational wave function is the most similar to this one has the largest transition moment and yields the most intense transition (is the easiest to reach). As the overlap of the vibrational wave function of a selected vibrational level of the excited state with the vibrational wave function of the initial state decreases, the transition moment into it decreases cf. Equation (1.36). Absorption intensity is proportional to the square of the overlap of the two nuclear wave functions, and drops to zero if they are orthogonal. This statement is known as the Franck-Condon principle (Franck, 1926 Condon, 1928 cf. also Schwartz, 1973) ... 

Figure 13. Real part of the overlap between the exact excited state nuclear wave function and the approximate wave function of the locally quadratic theory.

If 3 r becomes much larger than Gg and G, the diagonal energies of matrix (10) represent a better approximation. The wave-fimctions are not localized in the minima anymore because the ene etic (near-)degeneracy of the vibronic triplet is lifted. The finite overlap of the nuclear wave-functions xi demands that the electronic parts of the functions [Eq. (IS)] have to be orthogonal. The new eigenfimctions are the following linear combinations of 2> x> =... 

In the usual situation the overlap of the nuclear wave functions... 

Nevertheless, the classification of P transitions according to ft values can only be made roughly. For the exact classification one must start from the interaction matrix element Hfl given in O Eq. (2.75), which involves the overlap integral of initial and final nuclear wave functions and the power expansions of and ip (O Eqs. (2.76) and (2.77)). 

The diagonal elements Yhi hi m = are the populations on the electronic wave function in the electronic state. The off-diagonal elements represent the so-called coherences between the electronic states. To use the same notations as in ref. 24 and ref 28 we introduce the nuclear overlap/phase function J n by rewriting c]... 

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